Since \(\mathbb {C}^\times \cong \mathbb {R}\times \mathbb {T}\) , we can consider the quotient \(G/\mathbb {T}\) , which is \(\mathbb {R}\) -linear being nilpotent and simply connected [1]}. By a version of Proposition REF  for real Lie groups, there exists a length function exponentially distorted on the copy of \(\mathbb {R}\)  contained in \(\mathbb {C}^\times \) . (The proof of Proposition REF  based on results in [2]}, which hold not only for \(\mathbb {C}\)  but also for \(\mathbb {R}\) . We even do need the real version of [3]} stated in [4]} because use here a finite-dimensional representation.) Therefore this length function is (not strictly) exponentially distorted on \(\mathbb {C}^\times \) .
